First, it's interesting to see the effects of large volcanic eruptions on ocean heat content, leading to clear, smoother and noticeable changes in ocean heat content (OHC) for the 0-700 m region. Ocean warming really is the best way to measure a planet's energy imbalance.

Is the OHC of the 0-700 m region still increasing exponentially? Or is it now linear?

While I don't have all of that paper's data handy, I thought I could easily do something similar with the NOAA data I do have. In the following plot, I graphed the doubling time for heat content for the 0-700 meter region:

So, for example, the 0-700 m heat content the last quarter of 2015 (last data point) has doubled from 13.3 years ago.

In 1995, it had doubled from 10 years earlier. In 2010 it was double that of about 11 years earlier. Etc.

So the doubling time isn't quite constant -- it's increasing slowly with time. If it was constant, the change would be exponential -- as would its rate of change -- which is must faster than a second-order polynomial which gives a constant acceleration.